Question
Answer
$$((x+1)^2+(y-2)^2)(y-2)-(1+(\dfrac{y-2}{x+1})^2(x+1)^2)(2y-4)=x^2y+y^3-2x^2+2xy-6y^2-4x+13y-10-(1+(\dfrac{y-2}{x+1})^2(x^2+2x+1))(2y-4)$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}((x+1)^2+(y-2)^2)(y-2)-(1+(\frac{y-2}{x+1})^2(x+1)^2)(2y-4)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x^2+2x+1+y^2-4y+4)(y-2)-(1+(\frac{y-2}{x+1})^2(x^2+2x+1))(2y-4) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(x^2+y^2+2x-4y+5)(y-2)-(1+(\frac{y-2}{x+1})^2(x^2+2x+1))(2y-4) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}x^2y+y^3-2x^2+2xy-6y^2-4x+13y-10-(1+(\frac{y-2}{x+1})^2(x^2+2x+1))(2y-4)\end{aligned} $$ | |
| ① | Find $ \left(x+1\right)^2 $ using formula. $$ (A + B)^2 = \color{blue}{A^2} + 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ x } $ and $ B = \color{red}{ 1 }$. $$ \begin{aligned}\left(x+1\right)^2 = \color{blue}{x^2} +2 \cdot x \cdot 1 + \color{red}{1^2} = x^2+2x+1\end{aligned} $$Find $ \left(y-2\right)^2 $ using formula. $$ (A - B)^2 = \color{blue}{A^2} - 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ y } $ and $ B = \color{red}{ 2 }$. $$ \begin{aligned}\left(y-2\right)^2 = \color{blue}{y^2} -2 \cdot y \cdot 2 + \color{red}{2^2} = y^2-4y+4\end{aligned} $$Find $ \left(x+1\right)^2 $ using formula. $$ (A + B)^2 = \color{blue}{A^2} + 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ x } $ and $ B = \color{red}{ 1 }$. $$ \begin{aligned}\left(x+1\right)^2 = \color{blue}{x^2} +2 \cdot x \cdot 1 + \color{red}{1^2} = x^2+2x+1\end{aligned} $$ |
| ② | Combine like terms: $$ x^2+2x+ \color{blue}{1} +y^2-4y+ \color{blue}{4} = x^2+y^2+2x-4y+ \color{blue}{5} $$ |
| ③ | Multiply each term of $ \left( \color{blue}{x^2+y^2+2x-4y+5}\right) $ by each term in $ \left( y-2\right) $. $$ \left( \color{blue}{x^2+y^2+2x-4y+5}\right) \cdot \left( y-2\right) = x^2y-2x^2+y^3-2y^2+2xy-4x-4y^2+8y+5y-10 $$ |
| ④ | Combine like terms: $$ x^2y-2x^2+y^3 \color{blue}{-2y^2} +2xy-4x \color{blue}{-4y^2} + \color{red}{8y} + \color{red}{5y} -10 = x^2y+y^3-2x^2+2xy \color{blue}{-6y^2} -4x+ \color{red}{13y} -10 $$ |
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